670 



HISTORY OF THE THEORY OF NUMBERS. 



[CHAP. XXII 



A = m 2 +mn+n 2 , J3 = s(ra+tt) 3 +w 3 r+n 3 (r+s), o: = r 2 +rs+s 2 . Equate it to 

 the square of A-}-vBfAav 2 to get v rationally. 



Several 310 found 7 numbers in arithmetical progression the sum of whose 

 cubes is a biquadrate. Let nx 3x, nx 2x, -, nx-{-3x be the numbers. 

 Equating the sum of their cubes 7n 3 x 3 +84nx 3 to ra 4 z 4 , we get x. Or use 

 x, - , 7x, the sum of whose cubes is 784o; 3 . 



To find a rectangular parallelepiped whose edges, sum of edges, and 

 sum of faces, are rational squares, several 311 took x 2 , y 2 , z 2 as the adjacent 

 edges, and x 2 -\-y 2 +z 2 =(x+y z} 2 , whence z = xy/(x+y). Then 



if x 2 +xy+y 2 = D = (rx y) 2 , which gives x/y. C. Wilder took S = 

 [printed S = 4m 2 ], and z 2 = ?m/z(2-a 2 )/(2a). Then 



if 



= D 



m = 



6(2+a 2 )' 



Eliminating m from the assumed expression for x 2 , we get y in terms of x, 

 z, a, 6, which are arbitrary. [The solution is false as it satisfies neither 

 of the proposed equations, but only the combination of them which was 

 employed.] 



To find three positive integers the sum of any two of which is a square 

 and double the sum of all three is a biquadrate, R. Maffett and D. Robarts 312 

 took a 2 , b 2 , c 2 as the sums by pairs. Then shall a 2 +& 2 +c 2 be a biquadrate. 

 Take a = 3(p 2 +r 2 ), 6 = 4(p 2 -r 2 ), c = 8pr. Then Sa 2 = (5p 2 +5r 2 ) 2 , which 

 equals (25r 2 ) 2 for p = 2r. 



To find two integers whose sum, sum of squares, and sum of cubes, 

 are all squares, and sum of biquadrates is a cube, J. Whitley 313 used the 

 numbers x = 2rs, y = r 2 s 2 , whence x 2 xy +y 2 = D if r = 4s. Call X, Y the 

 products of x, y by 23 = 8+15. Then Z = 23-8s 2 , Y =15 -23s 2 satisfy the 

 first three conditions. Also X 4 +7 4 = 23 3 fc 8 , where Z = 23(8 4 +15 4 ), will be 

 a cube if s = t. C. Gill used x = b sin A, y b cos A with the sum a 2 . Then 



x z -{-yZ a?b 2 (l sin A cos A) = c 2 



if c = ab(l | sin A), cot %A = 4, whence x = 86/17, y = l5b/17. By their 

 sum, & = 17a 2 /23. The fourth condition is satisfied if a = 23 2 -54721. 

 E. Lucas 3130 proved that 2v 2 u 2 = w*, 2v 2 +u 2 = 3z 2 imply 



u z = v 2 = w 2 = z 2 = 1 . 



E. Lionnet 314 desired a number N which, as well as its biquadrate, is 

 the sum of the squares of two consecutive integers. J. Lissengon wrote 



310 The Gentleman's Diary, or Math. Repository, London, No. 76, 1816, 39, Quest. 1043. 



811 The Math. Diary, New York, 1, 1825, 125-7. 



812 Ladies' Diary, 1833, 35, Quest. 1542. 



313 The Lady's and Gentleman's Diary, London, 1854, 52-3, Quest. 1857. 

 3130 Nouv. Ann. Math., (2), 16, 1877, 414. 



311 Nouv. Ann. Math., (2), 19, 1880, 472-3. Repeated in Zeitschr. Math. Naturw. Unterricht, 



12, 1881, 268. 



